problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 8
values | synthetic bool 1
class | __index_level_0__ int64 0 742k |
|---|---|---|---|---|---|---|---|---|---|
XVI OM - II - Task 6
Prove that there does not exist a polyhedron whose every planar section is a triangle. | Every vertex of a polyhedron is the common point of at least three of its edges. Let $ A $ be a vertex of the polyhedron, and segments $ AB $ and $ AC $ as well as $ AB $ and $ AD $ be edges of two faces. Let us choose points $ P $ and $ Q $ on segments $ AC $ and $ AD $, respectively, different from the endpoints of t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 755 |
XXXII - I - Problem 10
Determine all functions $ f $ mapping the set of all rational numbers $ \mathbb{Q} $ to itself that satisfy the following conditions:
a) $ f(1)=2 $,
b) $ f(xy) = f(x)f(y)-f(x+y)+1 $ for $ x, y \in \mathbb{Q} $. | If the function $ f $ satisfies conditions a) and b), then the function $ g: \mathbb{Q} \to \mathbb{Q} $ defined by the formula $ g(x) = f(x)- 1 $ satisfies the conditions
a') $ g(1) = 1 $
and $ g(xy)+1 = (g(x)+1)(g(y)+1)-g(x+y)-1+1 $, $ g(xy)+1 = g(x)g(y)+g(x)+g(y)+1-g(x+y) $, i.e.
b') $ g(xy)+g(x+y) = g(x)g(y)+g(x)... | f(x)=x+1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 757 |
LVIII OM - II - Problem 1
The polynomial $ P(x) $ has integer coefficients. Prove that if the polynomials $ P(x) $ and $ P(P(P(x))) $ have a common real root, then they also have a common integer root. | Let the real number $ a $ be a common root of the polynomials $ P(x) $ and $ P(P(P(x))) $. From the equalities $ P(a)=0 $ and $ P(P(P(a)))=0 $, we obtain $ P(P(0))=0 $. The integer $ m=P(0) $ thus satisfies the conditions
which means that $ m $ is the desired common integer root. | proof | Algebra | proof | Yes | Yes | olympiads | false | 759 |
XLII OM - III - Problem 5
Non-intersecting circles $ k_1 $ and $ k_2 $ lie one outside the other. The common tangents to these circles intersect the line determined by their centers at points $ A $ and $ B $. Let $ P $ be any point on circle $ k_1 $. Prove that there exists a diameter of circle $ k_2 $, one end of whi... | Let the centers of circles $k_1$ and $k_2$ be denoted by $O_1$ and $O_2$, and their radii by $r_1$ and $r_2$. Points $A$ and $B$ lie on the line $O_1O_2$, and it does not matter which letter denotes which point. For the sake of clarity, let us agree that point $B$ lies on the segment $O_1O_2$, and point $A$ lies outsid... | proof | Geometry | proof | Yes | Yes | olympiads | false | 761 |
XII OM - II - Task 4
Find the last four digits of the number $ 5^{5555} $. | \spos{1} We will calculate a few consecutive powers of the number $ 5 $ starting from $ 5^4 $:
It turned out that $ 5^8 $ has the same last four digits as $ 5^4 $, and therefore the same applies to the numbers $ 5^9 $ and $ 5^5 $, etc., i.e., starting from $ 5^4 $, two powers of the number $ 5 $, whose exponents ... | 8125 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 766 |
XLV OM - I - Problem 6
The function $ f: \mathbb{R} \to \mathbb{R} $ is continuous. Prove that if for every real number $ x $ there exists a natural number $ n $ such that $ (\underbrace{f \circ \ldots \circ f}_{n})(x) = 1 $, then $ f(1)=1 $. | We will denote by $ f^r $ the $ r $-th iterate of the function $ f $, that is, the $ r $-fold composition (superposition) $ f \circ f \circ \ldots \circ f $.
Suppose, contrary to the thesis, that $ f(1) \neq 1 $. Let $ m $ be the smallest positive integer for which the equality
holds (such a number exists by the condi... | proof | Algebra | proof | Yes | Yes | olympiads | false | 767 |
XLVII OM - III - Problem 1
Determine all pairs $ (n,r) $, where $ n $ is a positive integer and $ r $ is a real number, for which the polynomial $ (x + 1)^n - r $ is divisible by the polynomial $ 2x^2 + 2x + 1 $. | Let $ Q_n(x) $ and $ R_n(x) $ denote the quotient and remainder, respectively, when the polynomial $ (x + 1)^n $ is divided by $ 2x^2 + 2x + 1 $. The pair $ (n,r) $ is one of the sought pairs if and only if $ R_n(x) $ is a constant polynomial, identically equal to $ r $.
For $ n = 1,2,3,4 $ we have:
Therefore,
... | (4k,(-\frac{1}{4})^k) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 770 |
XVII OM - I - Problem 6
Prove the theorem: If the coefficients $ a, b, c, d $ of the cubic equation $ ax^3 + bx^2 + cx + d = 0 $ are integers, where the number $ ad $ is odd and the number $ bc $ is even, then the equation cannot have three rational roots. | Suppose the given equation has three rational roots $x_1, x_2, x_3$. Let us transform the equation by introducing a new variable $y = ax$, i.e., substituting $\frac{y}{a}$ for $x$ in the equation.
We obtain the equation
whose roots are $y_1 = ax_1$, $y_2 = ax_2$, $y_3 = ax_3$. These numbers, being the products of... | proof | Algebra | proof | Yes | Yes | olympiads | false | 772 |
LV OM - I - Task 1
Given is a polygon with sides of rational length, in which all internal angles are equal to $ 90^{\circ} $ or $ 270^{\circ} $. From a fixed vertex, we emit a light ray into the interior of the polygon in the direction of the angle bisector of the internal angle at that vertex. The ray reflects accor... | Since every internal angle of the considered polygon $ \mathcal{W} $ is $ 90^{\circ} $ or $ 270^{\circ} $, all sides of this polygon (after an appropriate rotation) are aligned horizontally or vertically (Fig. 1). Let $ p_1/q_1, p_2/q_2, \ldots, p_n/q_n $ be the lengths of the consecutive sides of the polygon $ \mathca... | proof | Geometry | proof | Yes | Yes | olympiads | false | 773 |
XLIX OM - I - Problem 7
Determine whether there exists a convex polyhedron with $ k $ edges and a plane not passing through any of its vertices and intersecting $ r $ edges, such that $ 3r > 2k $. | Let $ W $ be any convex polyhedron; let $ k $ be the number of its edges, and $ s $ -- the number of faces. The boundary of each face contains at least three edges, and each edge is a common side of exactly two faces. Therefore, $ 2k \geq 3s $.
If a plane not passing through any vertex intersects $ r $ edges, then the ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 774 |
IX OM - I - Problem 12
Given a circle with radius $ r $ and rays $ AB $ and $ AC $ tangent to this circle at points $ B $ and $ C $. Determine the tangent to the given circle such that the segment contained within the angle $ BAC $ is the shortest. | The desired straight line should of course be sought among the tangents to the given circle that intersect the segments $AB$ and $AC$, because for other tangents, the segments contained in the angle $BAC$ are larger than the segments of the tangents parallel to them. Let $MN$ (Fig. 13) be the segment of the tangent lin... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 775 |
XXXVIII OM - II - Zadanie 4
Wyznaczyć wszystkie pary liczb rzeczywistych $ a, b $, dla których wielomiany $ x^4 + 2ax^2 + 4bx + a^2 $ i $ x^3 + ax - b $ mają dwa różne wspólne pierwiastki rzeczywiste.
|
Oznaczmy rozważane wielomiany przez $ P $ i $ Q $:
Załóżmy, że wielomiany $ P $ i $ Q $ mają wspólne pierwiastki rzeczywiste $ x_1 $, $ x_2 $ ($ x_1\neq x_2 $). Liczby $ x_1 $ $ x_2 $ są wówczas także pierwiastkami wielomianu
Stąd $ a \neq 0 $ i $ \Delta = 9b^2--4a^3 > 0 $.
Zgodnie z wzorami Viete'a... | (,0),where0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 779 |
XXXIV OM - III - Problem 5
In the plane, vectors $ \overrightarrow{a_1}, \overrightarrow{a_2}, \overrightarrow{a_3} $ of length 1 are given. Prove that one can choose numbers $ \varepsilon_1, \varepsilon_2, \varepsilon_3 $ equal to 1 or 2, such that the length of the vector $ \varepsilon_1\overrightarrow{a_1} + \varep... | Let's start from an arbitrary point $0$ with vectors $\overrightarrow{a_1}$, $\overrightarrow{a_2}$, $\overrightarrow{a_3}$. They determine three lines intersecting at point $0$. Among the angles determined by pairs of these lines, there is an angle of measure $\alpha$ not greater than $\frac{\pi}{3}$. Suppose this is ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 782 |
XXV OM - II - Problem 3
Prove that the orthogonal projections of vertex $ D $ of the tetrahedron $ ABCD $ onto the bisecting planes of the internal and external dihedral angles at edges $ \overline{AB} $, $ \overline{BC} $, and $ \overline{CA} $ lie on the same plane. | The bisecting plane is a plane of symmetry of the dihedral angle. Therefore, the image $D'$ of the vertex $D$ in the symmetry relative to any of the considered bisecting planes lies in the plane $ABC$. It follows that if $P$ is the orthogonal projection of the point $D$ onto the bisecting plane, then $P$ is the midpoin... | proof | Geometry | proof | Yes | Yes | olympiads | false | 783 |
XXXV OM - II - Problem 2
On the sides of triangle $ABC$, we construct similar isosceles triangles: triangle $APB$ outside triangle $ABC$ ($AP = PB$), triangle $CQA$ outside triangle $ABC$ ($CQ = QA$), and triangle $CRB$ inside triangle $ABC$ ($CR = RB$). Prove that $APRQ$ is a parallelogram or that points $A, P, R, Q$... | Consider a similarity with a fixed point $C$ that transforms $B$ into $R$ (this is the composition of a rotation around $C$ by the angle $BCR$ and a homothety with center $C$ and scale equal to the ratio of the base length to the arm length in each of the constructed isosceles triangles). This similarity transforms $A$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 784 |
XIII OM - I - Problem 3
Prove that the perpendiculars dropped from the centers of the excircles of a triangle to the corresponding sides of the triangle intersect at one point. | A simple solution to the problem can be inferred from the observation that on each side of the triangle, the point of tangency of the inscribed circle and the point of tangency of the corresponding excircle are symmetric with respect to the midpoint of that side. For example, if the inscribed circle in triangle $ABC$ t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 788 |
XLV OM - III - Task 4
We have three unmarked vessels: an empty $ m $-liter, an empty $ n $-liter, and a full $(m+n)$-liter vessel of water. The numbers $ m $ and $ n $ are relatively prime natural numbers. Prove that for every number $ k \in \{1,2, \ldots , m+n-1\} $, it is possible to obtain exactly $ k $ liters of w... | When pouring water from one container to another, we either completely empty the container we are pouring from or completely fill the container we are pouring into. (Since the containers have no markings, this is the only method that allows us to control the volume of water being poured.) Thus, after each pour, we have... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 789 |
XLVI OM - II - Problem 3
Given are positive irrational numbers $ a $, $ b $, $ c $, $ d $, such that $ a+b = 1 $. Prove that $ c+d = 1 $ if and only if for every natural number $ n $ the equality $ [na] +[nb] = [nc] + [nd] $ holds.
Note: $ [x] $ is the greatest integer not greater than $ x $. | Let's take any natural number $ n \geq 1 $. According to the definition of the symbol $ [x] $,
The products $ na $, $ nb $, $ nc $, $ nd $ cannot be integers (since the numbers $ a $, $ b $, $ c $, $ d $ are irrational by assumption), and therefore in none of the listed relationships can equality hold. Addi... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 791 |
LIX OM - III - Task 5
The areas of all sections of the parallelepiped $ \mathcal{R} $ by planes passing through the midpoints of three of its edges,
none of which are parallel and do not have common points, are equal. Prove that the parallelepiped
$ \mathcal{R} $ is a rectangular parallelepiped. | Let's assume that the areas of all the sections mentioned in the problem are equal to $S$.
Let $ABCD$ and $EFGH$ be the bases of the parallelepiped $\mathcal{R}$ (Fig. 3), let $O$ be its center of symmetry, and let $I, J, K, L, M, N$ denote the midpoints of the edges $AE, EF, FG, GC, CD, DA$, respectively. Then the fol... | proof | Geometry | proof | Yes | Yes | olympiads | false | 792 |
LVIII OM - I - Problem 12
The polynomial $ W $ with real coefficients takes only positive values in the interval $ \langle a;b\rangle $ (where $ {a<b} $). Prove that there exist polynomials $ P $ and $ Q_1,Q_2,\ldots,Q_m $ such that
for every real number $ x $. | We will conduct the proof by induction on the degree of the polynomial $W$.
If $W(x) \equiv c$ is a constant polynomial, then of course $c > 0$ and in this case we can take $P(x) \equiv \sqrt{c}$, $m=1$, and $Q_1(x) \equiv 0$.
Now suppose that the thesis of the problem is true for all polynomials of degree less than $n... | proof | Algebra | proof | Yes | Yes | olympiads | false | 793 |
XXVII OM - II - Problem 6
Six points are placed on a plane in such a way that any three of them are vertices of a triangle with sides of different lengths. Prove that the shortest side of one of these triangles is also the longest side of another one of them. | Let $P_1, P_2, \ldots, P_6$ be given points. In each of the triangles $P_iP_jP_k$, we paint the shortest side red. In this way, some segments $\overline{P_rP_s}$ are painted red, while others remain unpainted.
It suffices to prove that there exists a triangle with vertices at the given points, all of whose sides are pa... | proof | Geometry | proof | Yes | Yes | olympiads | false | 795 |
XV OM - I - Problem 9
Prove that the quotient of the sum of all natural divisors of an integer $ n > 1 $ by the number of these divisors is greater than $ \sqrt{n} $. | Let $ d_1, d_2, \ldots, d_s $ denote all the natural divisors of a given integer $ n > 1 $. Among these divisors, there are certainly unequal numbers, such as $ 1 $ and $ n $. According to the Cauchy inequality, the arithmetic mean of positive numbers that are not all equal is greater than the geometric mean of these n... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 796 |
XXXIV OM - I - Problem 7
Let $ S_n $ be the set of sequences of length $ n $ with terms $ -1 $, $ +1 $. We define the function $ f: S_n - \{(-1, 1, 1,\ldots, 1)\} \to S_n $ as follows: if $ (a_1, \ldots, a_n) \in S_n - \{(-1, 1, 1,\ldots, 1)\} $ and $ k = \max_{1\leq j \leq n} \{j \;:\; a_1\cdot a_2\cdot \ldots \cdot ... | To simplify the notation, we introduce the symbol $ f^{(k)} $ to denote the $ k $-fold iteration of the function $ f $:
The thesis of the problem will be proved by induction. The definition of the function $ f $ depends, of course, on the natural number $ n $. Where it helps to avoid misunderstandings, we will write $... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 799 |
XIX OM - I - Problem 7
Points $ D, E, F $ are the midpoints of the sides of triangle $ ABC $. Prove that if the circumcircles of triangles $ ABC $ and $ DEF $ are tangent, then the point of tangency is one of the points $ A $, $ B $, $ C $ and triangle $ ABC $ is a right triangle. | Suppose that the circle $c$ circumscribed around triangle $ABC$ and the circle $k$ circumscribed around triangle $DEF$ are tangent at point $T$ (Fig. 5).
Triangle $DEF$ lies inside circle $c$, so the tangency of circles $c$ and $k$ is internal, and the center $S$ of circle $k$ lies on the ray $TO$. Triangle $DEF$ is si... | proof | Geometry | proof | Yes | Yes | olympiads | false | 803 |
VIII OM - III - Task 5
Given a line $ m $ and a segment $ AB $ parallel to it. Divide the segment $ AB $ into three equal parts using only a ruler, i.e., by drawing only straight lines. | We first find the midpoint $ S $ of segment $ AB $ as in problem 17 (Fig. 16) (Fig. 23). We draw lines $ AT $ and $ SE $, which intersect at point $ G $, and line $ GB $ intersecting line $ m $ at point $ H $. Since $ AS = SB $ and $ m \parallel AB $, then $ DT = TE $ and $ TE = EH $. Therefore, if $ K $ is the point o... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 804 |
LIV OM - III - Task 1
In an acute triangle $ABC$, segment $CD$ is an altitude. Through the midpoint $M$ of side $AB$, a line is drawn intersecting rays $CA$ and $CB$ at points $K$ and $L$, respectively, such that $CK = CL$. Point $S$ is the center of the circumcircle of triangle $CKL$. Prove that $SD = SM$. | From Menelaus' theorem applied to triangle $ABC$ we get
Let $E$ be the second intersection point of line $CS$ with the circumcircle of triangle $CKL$ (Fig. 1). From the equality $CK = CL$, it follows that $EK = EL$. Moreover, $\measuredangle AKE = 90^\circ = \measuredangle BLE$.
Therefore, the right triangles $AK... | proof | Geometry | proof | Yes | Yes | olympiads | false | 805 |
XIX OM - I - Problem 2
Let $ p(n) $ denote the number of prime numbers not greater than the natural number $ n $. Prove that if $ n \geq 8 $, then $ p(n) < \frac{n}{2} $ | If $ n $ is an even number, then in the set of natural numbers from $ 1 $ to $ n $, there are $ \frac{n}{2}-1 $ even numbers different from $ 2 $, and thus composite, and at least one odd number, namely $ 1 $, which is not a prime number. Prime numbers belong to the set of the remaining $ \frac{n}{2} $ numbers, so $ p(... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 807 |
XXXVII OM - II - Problem 3
Let S be a sphere circumscribed around a regular tetrahedron with an edge length greater than 1. The sphere $ S $ is represented as the union of four sets. Prove that there exists one of these sets that contains points $ P $, $ Q $, such that the length of the segment $ PQ $ exceeds 1. | We will first prove a lemma that is a one-dimensional version of this theorem.
Lemma. The circle $\omega$ circumscribed around an equilateral triangle with a side length greater than $1$ is represented as the union of three sets: $\omega = U \cup V \cup W$. Then, there exist points $P$, $Q$ in one of the sets $U$, $V$,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 808 |
X OM - III - Task 6
Given is a triangle in which the sides $ a $, $ b $, $ c $ form an arithmetic progression, and the angles also form an arithmetic progression. Find the ratios of the sides of this triangle. | Suppose that triangle $ABC$ satisfies the conditions of the problem, with $A \leq B \leq C$, and thus $a \leq b \leq c$. In this case
From equation (1) and the equation $A + B + C = \pi$, it follows that $B = \frac{\pi}{3}$; the cosine rule then gives the relationship
Eliminating $b$ from equations (2... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 809 |
XXII OM - II - Problem 1
In how many ways can $ k $ fields of a chessboard $ n \times n $ ($ k \leq n $) be chosen so that no two of the selected fields lie in the same row or column? | First, one can choose $ k $ rows in which the selected fields will lie. This can be done in $ \displaystyle \binom{n}{k} $ ways. Then, in each of these rows, one must sequentially choose one of $ n $ fields, one of $ n - 1 $ fields lying in the remaining columns, etc. - one of $ n - k + 1 $ fields. Therefore, the total... | \binom{n}{k}n(n-1)\ldots(n-k+1) | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 812 |
XXXIII OM - III - Problem 1
Indicate such a way of arranging $ n $ girls and $ n $ boys around a round table so that the number $ d_n - c_n $ is maximized, where $ d_n $ is the number of girls sitting between two boys, and $ c_n $ is the number of boys sitting between two girls. | Consider any arrangement of $ n $ girls and $ n $ boys. Let's call a group of $ k $ girls (for $ k > 1 $) any sequence of $ k $ girls sitting next to each other, if the first and the $ k $-th girl in this sequence are seated next to a boy at the table. Let $ D_n $ be the number of groups of girls in a given arrangement... | [\frac{n}{2}]-1 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 814 |
IX OM - I - Problem 11
Prove that if two quadrilaterals have the same midpoints of their sides, then they have equal areas. Show the validity of an analogous theorem for convex polygons with any even number of sides. | Let $M$, $N$, $P$, $Q$ be the midpoints of the sides $AB$, $BC$, $CD$, $DA$ of quadrilateral $ABCD$ (Fig. 10). By the theorem on the segment joining the midpoints of two sides of a triangle, in triangle $ABD$ we have
and in triangle $BCD$ we have
thus $MQ = NP$ and $MQ \parallel NP$, so quadrilateral $MNPQ$ is a para... | proof | Geometry | proof | Yes | Yes | olympiads | false | 815 |
XXIV OM - II - Problem 2
In a given square, there are nine points, no three of which are collinear. Prove that three of them are vertices of a triangle with an area not exceeding $ \frac{1}{8} $ of the area of the square. | Let's first prove the
Lemma. If triangle $ABC$ is contained within a certain rectangle, then the area of the triangle is not greater than half the area of that rectangle.
Proof. Let $A$, $B$, $C$ be the projections of the vertices of triangle $ABC$ onto one of the sides of the rectangle $PQRS$ containing this triangle ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 817 |
LI OM - II - Task 2
The bisector of angle $ BAC $ of triangle $ ABC $ intersects the circumcircle of this triangle at point $ D $ different from $ A $. Points $ K $ and $ L $ are the orthogonal projections of points $ B $ and $ C $, respectively, onto the line $ AD $. Prove that | Let $ o $ be the circumcircle of triangle $ ABC $. From the equality of angles $ BAD $ and $ DAC $, it follows that the lengths of arcs $ BD $ and $ DC $ of circle $ o $ are equal (the length of arc $ XY $ is measured from point $ X $ to point $ Y $ in the counterclockwise direction).
om51_2r_img_1.jpg
om51_2r_img_2.jp... | proof | Geometry | proof | Yes | Yes | olympiads | false | 819 |
XII OM - II - Task 3
Prove that for any angles $ x $, $ y $, $ z $ the following equality holds | We transform the right side of equation (1) by first applying the formula $ 2 \sin \alpha \sin \beta = \cos (\alpha - \beta) - \cos (\alpha + \beta) $, and then the formulas for $ \cos (\alpha + \beta) $ and $ \cos (\alpha - \beta) $:
After substituting in the last expression $ \sin^2 x \sin^2 y = (1 - \cos^2 x) ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 820 |
XXVII OM - I - Problem 11
Transmitting and receiving stations are sequentially connected: $ S_0 $ with $ S_1 $, $ S_1 $ with $ S_2 $, ..., $ S_n $ with $ S_{n+1} $, ... Station $ S_0 $ transmits a signal of 1 or -1, station $ S_1 $ receives the same signal with probability $ 1 - \varepsilon $, and the opposite signal ... | We have of course $ p^n(1 \mid 1) = p^n(-1 \mid -1) $, $ p^n(1 \mid 1) + p^n(1 \mid -1) = 1 $ and $ p^n(-1 \mid 1) + p^n(-1 \mid -1) = 1 $. Therefore, $ p^n(1 \mid -1) = p^n(-1 \mid 1) $.
From the above equalities, it follows that the limits of the sequences $(p^n(1 \mid 1))$ and $(p^n(-1 \mid -1))$ are equal, and simi... | Algebra | math-word-problem | Yes | Yes | olympiads | false | 821 | |
XXX OM - II - Task 3
In space, a line $ k $ and a cube with vertex $ M $ and edges $ \overline{MA} $, $ \overline{MB} $, $ \overline{MC} $, each of length 1, are given. Prove that the length of the orthogonal projection of the edge $ MA $ onto the line $ k $ is equal to the area of the orthogonal projection of the squ... | Without loss of generality, we can assume that the line $k$ passes through the point $M$ (Fig. 12) and that the plane $\pi$ perpendicular to the line $k$ also contains the point $M$. Let $A$ be the orthogonal projection of the point $A$ onto the line $k$.
om30_2r_img_12.jpg
As is known, if two planes intersect at an an... | proof | Geometry | proof | Yes | Yes | olympiads | false | 822 |
III OM - I - Task 4
a) Given points $ A $, $ B $, $ C $ not lying on a straight line. Determine three mutually parallel lines passing through points $ A $, $ B $, $ C $, respectively, so that the distances between adjacent parallel lines are equal.
b) Given points $ A $, $ B $, $ C $, $ D $ not lying on a plane. Deter... | a) Suppose that the lines $a$, $b$, $c$ passing through points $A$, $B$, $C$ respectively and being mutually parallel satisfy the condition of the problem, that is, the distances between adjacent parallel lines are equal. Then the line among $a$, $b$, $c$ that lies between the other two is equidistant from them. Let th... | 12 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 823 |
LV OM - I - Task 2
Determine whether there exists a prime number $ p $ and non-negative integers $ x $, $ y $, $ z $ satisfying the equation | Suppose there exist numbers $ p $, $ x $, $ y $, $ z $ satisfying the given equation in the problem. Since $ p $ is a prime number, the divisors of $ p^z $ are only powers of $ p $. Therefore, there exist non-negative integers $ a $, $ b $ such that $ 12x + 5 = p^a $ and $ 12y + 7 = p^b $. From this, it follows that th... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 826 |
LVII OM - III - Problem 4
We perform the following operation on a triplet of numbers. We select two of these numbers and replace them with their sum and their product, while the remaining number remains unchanged. Determine whether, starting from the triplet (3,4,5) and performing this operation, we can again obtain a... | If in the first step we choose the numbers 3 and 5, we will get the triplet (4, 8, 15), in which exactly one number is odd. Choosing two even numbers from such a triplet, we replace them with even numbers, whereas deciding on one even and one odd number, we get numbers of different parities. Therefore, from a triplet i... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 827 |
XLVI OM - I - Problem 8
In a regular $ n $-sided pyramid, the angles of inclination of the lateral face and the lateral edge to the base plane are $ \alpha $ and $ \beta $, respectively. Prove that $ \sin^2 \alpha - \sin^2 \beta \leq \tan ^2 \frac{\pi}{2n} $. | Let's assume that the side of the regular $ n $-gon $ A_1A_2\ldots A_n $, which serves as the base of the pyramid, has a length of $ 2a $. Let $ O $ be the common center of the circle circumscribed around this polygon and the circle inscribed in it; let the radii of these circles be (respectively) $ R $ and $ r $. Deno... | proof | Geometry | proof | Yes | Yes | olympiads | false | 828 |
XXV OM - III - Problem 1
In the tetrahedron $ ABCD $, the edge $ \overline{AB} $ is perpendicular to the edge $ \overline{CD} $ and $ \measuredangle ACB = \measuredangle ADB $. Prove that the plane determined by the edge $ \overline{AB} $ and the midpoint of the edge $ \overline{CD} $ is perpendicular to the edge $ \o... | We will first prove the
Lemma. If lines $AB$ and $PQ$ intersect at a right angle at point $P$, then the number of points on the ray $PQ^\to$ from which the segment $\overline{AB}$ is seen at an angle $\alpha$ is $0$, $1$, or $2$.
Proof. As is known, the set of points contained in the half-plane with edge $AB$ and conta... | proof | Geometry | proof | Yes | Yes | olympiads | false | 831 |
XXIII OM - I - Problem 6
Determine for which digits $ a $ the decimal representation of a number $ \frac{n(n+1)}{2} $ ($ n\in \mathbb{N} $) consists entirely of the digit $ a $. | Of course, $ a $ cannot be zero. If the decimal representation of the number
$ \displaystyle t_n = \frac{1}{2} n (n + 1) $ consists of $ k $ ones, where $ k \geq 2 $, then $ 9 t_n = 10^k - 1 $. From this, after simple transformations, we obtain $ (3n + 1) \cdot (3n + 2) = 2 \cdot 10^k = 2^{k+1} \cdot 5^k $. Since the c... | 5,6 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 832 |
XXXVIII OM - III - Problem 2
A regular $ n $-gon is inscribed in a circle of length 1. From among the arcs of the circle having endpoints at the vertices of this polygon, we randomly draw five arcs $ L_1, \ldots, L_5 $ with replacement, where the probability of drawing each arc is the same. Prove that the expected val... | The vertices of a polygon divide a circle into $n$ arcs of length $1/n$. Let us number these arcs: $J_1, \ldots, J_n$. Denote by $\mathcal{L}$ the set from which the selection is made. The elements of the set $\mathcal{L}$ are arcs of a given circle, each of which has two ends, which are different vertices of the consi... | \frac{1}{32} | Combinatorics | proof | Yes | Yes | olympiads | false | 833 |
XXXIX OM - III - Problem 3
Let $ W $ be a polygon (not necessarily convex) with a center of symmetry. Prove that there exists a parallelogram containing $ W $ such that the midpoints of the sides of this parallelogram lie on the boundary of $ W $. | Among all triangles $OAB$, where $A$, $B$ are vertices of the polygon $W$, and $O$ is its center of symmetry, let us choose the triangle with the maximum area (there are at least four such triangles, and there may be more; we choose any one of them). Let $OMN$ be the selected triangle, and let $M$, $N$ be the vertices ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 834 |
XX OM - II - Task 6
Prove that every polyhedron has at least two faces with the same number of sides. | Let $ s $ denote the number of faces of the polyhedron $ W $, and $ P $ be the face with the largest number of sides, which we will denote by $ n $. The face $ P $ is adjacent along its sides to $ n $ other (pairwise distinct) faces of the polyhedron, so $ s \geq n+1 $. On the other hand, the number of sides of each fa... | proof | Geometry | proof | Yes | Yes | olympiads | false | 836 |
XLIX OM - I - Problem 12
Let $ g(k) $ be the greatest prime divisor of the integer $ k $, when $ |k|\geq 2 $, and let $ g(-1) = g(0) = g(1) = 1 $. Determine whether there exists a polynomial $ W $ of positive degree with integer coefficients, for which the set of numbers of the form $ g(W(x)) $ ($ x $ - integer) is fi... | We will prove that there does not exist a polynomial with the given property. For a proof by contradiction, suppose that
is a polynomial of degree $ n \geq 1 $ with integer coefficients and that the set of numbers of the form $ g(W(x)) $ (for integer values of $ x $) is finite. This means: there exists a natural numbe... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 837 |
LVII OM - I - Problem 3
An acute triangle $ABC$ is inscribed in a circle with center $O$. Point $D$ is the orthogonal projection of point $C$ onto line $AB$, and points $E$ and $F$ are the orthogonal projections of point $D$ onto lines $AC$ and $BC$, respectively. Prove that the area of quadrilateral $EOFC$ is equal t... | Let $ P $ be the point symmetric to point $ C $ with respect to point $ O $ (Fig. 1). Since triangle $ ABC $ is acute,
points $ A $, $ P $, $ B $, and $ C $ lie on a circle with center $ O $ in this exact order.
om57_1r_img_1.jpg
The areas of triangles $ COE $ and $ POE $ are equal, as these triangles have a common he... | proof | Geometry | proof | Yes | Yes | olympiads | false | 839 |
XL OM - I - Task 4
Prove that it is impossible to cut a square along a finite number of segments and arcs of circles in such a way that the resulting pieces can be assembled into a circle (pieces can be flipped). | Suppose the division mentioned in the task is feasible. The resulting parts of the square are denoted by $K_1, \ldots, K_n$. The boundary of each figure $K_i$ consists of a finite number of line segments and arcs of circles. Among the arcs of circles that are parts of the boundary of $K_i$, some are convex outward from... | proof | Geometry | proof | Yes | Yes | olympiads | false | 841 |
XXVI - I - Problem 11
A ship is moving at a constant speed along a straight line with an east-west direction. Every $ T $ minutes, the direction of movement is randomly chosen: with probability $ p $, the ship moves in the eastern direction for the next $ T $ minutes, and with probability $ q= 1-p $, it moves in the w... | Let's assume that a submarine fires a torpedo at the moment when a lottery is taking place on the ship. In the case of firing the torpedo at a different time, the reasoning proceeds similarly. We can also assume, without loss of generality, that $ p \geq q $, i.e., $ \displaystyle p \geq \frac{1}{2} $. After $ T $ minu... | \frac{4}{9} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 842 |
XIII OM - I - Problem 11
Given is a quadrilateral $ABCD$ whose diagonals intersect at a right angle at point $M$. Prove that the $8$ points where perpendiculars drawn from point $M$ to the lines $AB$, $BC$, $CD$, and $DA$ intersect the sides of the quadrilateral lie on a circle. | Let $P$, $Q$, $R$, $S$ denote the feet of the perpendiculars dropped from point $M$ to the sides $AB$, $BC$, $CD$, $DA$ respectively (Fig. 16).
First, we will prove that the points $P$, $Q$, $R$, $S$ lie on a circle by showing that the sum of two opposite angles of quadrilateral $PQRS$ equals $180^\circ$.
Each of the q... | proof | Geometry | proof | Yes | Yes | olympiads | false | 843 |
XV OM - III - Task 4
Prove that if the roots of the equation $ x^3 + ax^2 + bx + c = 0 $, with real coefficients, are real, then the roots of the equation $ 3x^2 + 2ax + b = 0 $ are also real. | The task boils down to showing that the given assumptions imply the inequality
Let $ x_1 $, $ x_2 $, $ x_3 $ denote the roots of equation (2); according to the assumption, they are real numbers.
We know that
Hence
Note. Using elementary knowledge of derivatives, the problem can be solved much more simply. It i... | proof | Algebra | proof | Yes | Yes | olympiads | false | 844 |
LVIII OM - III - Task 2
A positive integer will be called white if it is equal to 1
or is the product of an even number of prime numbers (not necessarily distinct).
The remaining positive integers will be called black.
Determine whether there exists a positive integer such that the sum of its white
divisors is equal ... | For an integer $ k>1 $, let us denote
We will prove that for any relatively prime positive integers $ l $, $ m $, the equality
is true.
Indeed, every positive divisor $ d $ of the product $ lm $ has a unique representation in the form $ d=ab $, where $ a $ is a divisor of the number $ l $, and $ b $ is a ... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 846 |
XLIV OM - II - Problem 5
On the sides $ BC $, $ CA $, $ AB $ of triangle $ ABC $, points $ D $, $ E $, and $ F $ are chosen respectively such that the incircles of triangles $ AEF $, $ BFD $, and $ CDE $ have radii equal to $ r_1 $. The incircles of triangles $ DEF $ and $ ABC $ have radii $ r_2 $ and $ r $, respectiv... | Let $ P $, $ Q $, $ R $ be the points of tangency of the incircle of triangle $ ABC $ with the sides $ BC $, $ CA $, $ AB $, respectively, and let $ K $ be the point of tangency of the incircle of triangle $ EAF $ with the side $ AF $. The centers of these circles are denoted by $ I $ and $ J $, respectively (see Figur... | r_1+r_2 | Geometry | proof | Yes | Yes | olympiads | false | 847 |
IV OM - II - Task 3
A triangular piece of sheet metal weighs $ 900 $ g. Prove that by cutting this sheet along a straight line passing through the center of gravity of the triangle, it is impossible to cut off a piece weighing less than $ 400 $ g. | We assume that the sheet is of uniform thickness everywhere; the weight of a piece of sheet is then proportional to the area of the plane figure represented by that piece of sheet. The task is to show that after cutting a triangle along a straight line passing through its center of gravity, i.e., through the point of i... | proof | Geometry | proof | Yes | Yes | olympiads | false | 850 |
XII OM - III - Problem 5
Four lines intersecting at six points form four triangles. Prove that the circumcircles of these triangles have a common point. | Four lines intersecting at $6$ points form a figure known as a complete quadrilateral (see Problems from Mathematical Olympiads, vol. I, Warsaw 1960. Problem 79). Let us denote these points by $A$, $B$, $C$, $D$, $E$, $F$ (Fig. 21) in such a way that on the given lines lie the respective triplets of points $(A, B, C)$,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 851 |
XXX OM - I - Task 7
For a given natural number $ n $, calculate the number of integers $ x $ in the interval $ [1, n] $ for which $ x^3 - x $ is divisible by $ n $. | If $ n = p^k $, where $ k \geq 1 $ and $ p $ is an odd prime, then for any integer $ x $, at most one of the numbers $ x - 1, x, x + 1 $ is divisible by $ p $. Therefore, the number $ x^3 - x = (x - 1) x(x + 1) $ is divisible by $ p^k $ if and only if one of the numbers $ x - 1, x, x + 1 $ is divisible by $ p^k $, whic... | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 853 | |
XLIII OM - II - Problem 1
Each vertex of a certain polygon has both coordinates as integers; the length of each side of this polygon is a natural number. Prove that the perimeter of the polygon is an even number. | Let $ A_1,\ldots,A_n $ be the consecutive vertices of a polygon; it will be convenient for us to denote the vertex $ A_n $ also by $ A_0 $. The boundary of the polygon forms a closed broken line, hence
Let us denote the coordinates of the $ i $-th vector of the above sum by $ u_i $, $ v_i $, and its length by $ w... | proof | Geometry | proof | Yes | Yes | olympiads | false | 854 |
XLII OM - III - Problem 2
Let $ X $ be the set of points in the plane $ (x.y) $ with both coordinates being integers. A path of length $ n $ is any sequence $ (P_0, P_1,\ldots, P_n) $ of points in the set $ X $ such that $ |P_{i-1}P_{i}|=1 $ for $ i \in \{1,2,\ldots,n\} $. Let $ F(n) $ be the number of different paths... | For any path of length $ n $, with the starting point $ P_0 = (0,0) $, we associate a sequence of $ 2n $ symbols, each of which is a zero or a one. We do this as follows. If the path has the form $ (P_0,P_1,\ldots,P_n) $, then the corresponding zero-one sequence
called the code of the path, is determined by the set of... | \binom{2n}{n} | Combinatorics | proof | Yes | Yes | olympiads | false | 855 |
XXIX OM - I - Problem 4
Let $ Y $ be a figure consisting of closed segments $ \overline{OA} $, $ \overline{OB} $, $ \overline{OC} $, where point O lies inside the triangle $ ABC $. Prove that in no square can one place infinitely many mutually disjoint isometric images of the figure $ Y $. | Let $ d $ be the smallest of the numbers $ OA $, $ OB $, $ OC $. Choose points $ P \in \overline{OA} $, $ Q \in \overline{OB} $, $ R \in \overline{OC} $ such that $ OP= OQ = OR= \frac{d}{2} $ (Fig. 6).
Then $ PQ < OP + OQ = d $ and similarly $ PR < d $ and $ QR < d $. If $ Y $ is a figure consisting of closed segments ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 858 |
XLV OM - I - Problem 5
Prove that if the polynomial $ x^3 + ax^2 + bx + c $ has three distinct real roots, then the polynomial $ x^3 + ax^2 + \frac{1}{4}(a^2 + b)x +\frac{1}{8}(ab - c) $ also has three distinct real roots. | Let's denote the polynomials in the problem by $P(x)$ and $Q(x)$:
Replacing the variable $x$ in the polynomial $P(x)$ with the difference $2x - a$ and transforming the obtained expression:
From the obtained identity:
it immediately follows that if a number $\xi$ is a root of the polynomial $P(x)$, then the number $-... | proof | Algebra | proof | Yes | Yes | olympiads | false | 859 |
LI OM - I - Task 5
Determine all pairs $ (a,b) $ of natural numbers for which the numbers $ a^3 + 6ab + 1 $ and $ b^3 + 6ab + 1 $ are cubes of natural numbers. | Let $ a $ and $ b $ be numbers satisfying the conditions of the problem. Without loss of generality, we can assume that $ a \leq b $. Then
Since the number $ b^3+6ab + 1 $ is a cube of an integer, the following equality must hold
Transforming the above relationship equivalently, we obtain step by step
... | (1,1) | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 861 |
IX OM - III - Task 1
Prove that the product of three consecutive natural numbers, of which the middle one is a cube of a natural number, is divisible by $ 504 $. | We need to prove that if $a$ is a natural number greater than $1$, then the number
is divisible by $504 = 7 \cdot 8 \cdot 9$. Since the numbers $7$, $8$, and $9$ are pairwise coprime, the task reduces to proving the divisibility of $N$ by each of these numbers.
a) The number $a$ can be represented in the form $a = 7k ... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 865 |
XXXV OM - I - Problem 9
Three events satisfy the conditions:
a) their probabilities are equal,
b) any two of them are independent,
c) they do not occur simultaneously.
Determine the maximum value of the probability of each of these events. | om35_1r_img_4.jpg
Suppose that $ A $, $ B $, $ C $ are these events, each with a probability of $ p $. Given the assumption of independence between any two of the considered events, the probability of each intersection $ A \cap B $, $ B \cap C $, $ C \cap A $ is $ p^2 $. Furthermore, $ A \cap B \cap C \ne 0 $. Therefor... | \frac{1}{2} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 866 |
XXXIV OM - III - Problem 6
Prove that if all dihedral angles of a tetrahedron are acute, then all its faces are acute triangles. | We assign a unit vector perpendicular to each face and directed outward from the tetrahedron. Consider two faces forming an angle $ \alpha $. The vectors assigned to these faces form an angle $ \pi - \alpha $ (Fig. 12). Therefore, for all dihedral angles of the tetrahedron to be acute, it is necessary and sufficient th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 867 |
LIX OM - III - Task 1
In the fields of an $ n \times n $ table, the numbers $ 1, 2, \dots , n^2 $ are written, with the numbers $ 1, 2, \dots , n $ being in the first row (from left to right), the numbers $ n +1, n +2, \dots ,2n $ in the second row, and so on.
$n$ fields of the table have been selected, such that no ... | We will first prove that
Let $ b_i $ denote the column number in which the number $ a_i $ is located. Then,
From the conditions of the problem, it follows that the sequence $ (b_1,b_2,\dots ,b_n) $ is a permutation of the sequence $ (1, 2,\dots ,n) $. Therefore,
and in view of formula (2), the proof ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 872 |
XVIII OM - III - Problem 3
In a room, there are 100 people, each of whom knows at least 67 others. Prove that there is a quartet of people in the room where every two people know each other. We assume that if person $A$ knows person $B$, then person $B$ also knows person $A$. | In the set $M$ of people in the room, there are at most $32$ people who do not know a certain person $A$ (assuming that person $A$ knows themselves).
If $A$, $B$, and $C$ are any three people in the set $M$, then there is a person $D$ in the set $M$, different from $A$, $B$, and $C$, who knows these three people. Indee... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 874 |
XLII OM - I - Problem 1
On the shore of a lake in the shape of a circle, there are four piers $K$, $L$, $P$, $Q$. A kayak departs from pier $K$ heading towards pier $Q$, and a boat departs from pier $L$ heading towards pier $P$. It is known that if, maintaining their speeds, the kayak headed towards pier $P$ and the b... | om42_1r_img_1.jpg
Suppose the kayak is moving at a speed of $ v_k $, and the boat is moving at a speed of $ v_l $. Let $ X $ be the point where a "collision" would occur. The lengths of the segments into which point $ X $ divides the chords $ KP $ and $ LQ $ are related by $ |KX| \cdot |PX| = |LX| \cdot |QX| $ (Figure ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 875 |
LIX OM - III - Task 6
Let $ S $ be the set of all positive integers that can be represented in the form $ a^2 + 5b^2 $
for some relatively prime integers $ a $ and $ b $. Furthermore, let $ p $ be a prime number that gives a remainder
of 3 when divided by 4. Prove that if some positive multiple of the number $ p $ bel... | Given the conditions of the problem, the number $a^2 + 5b^2$ is divisible by $p$ for some relatively prime integers $a$ and $b$.
The number $b$ is not divisible by $p$; otherwise, from the divisibility $p \mid b$ and $p \mid a^2 + 5b^2$, we would obtain $p \mid a$, contradicting the relative primality of $a$ and $b$. ... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 876 |
XXVII OM - II - Problem 3
We consider a hemispherical cap that does not contain any great circle. The distance between points $A$ and $B$ on such a cap is defined as the length of the arc of the great circle of the sphere with endpoints at points $A$ and $B$, which is contained in the cap. Prove that there is no isome... | Let a given spherical cap correspond to the central angle $ \alpha $. From the assumption, we have $ 0 < \alpha < \frac{\pi}{2} $. For any angle $ \beta $ satisfying $ 0 < \beta < \alpha $, consider a regular pyramid $ OABCD $, whose base is a square $ ABCD $ inscribed in the given spherical cap, and whose vertex $ O $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 877 |
XXVIII - I - Problem 8
Prove that the set $ \{1, 2, \ldots, 2^{s+1}\} $ can be partitioned into two $ 2s $-element sets $ \{x_1, x_2, \ldots, x_{2^s}\} $, $ \{y_1, y_2, \ldots, y_{2^s}\} $, such that for every natural number $ j \leq s $ the following equality holds | We will apply induction with respect to $s$. For $s = 1$, we have the set $\{1, 2, 3, 4\}$. By splitting it into subsets $\{1, 4\}$ and $\{2, 3\}$, we will have the conditions of the problem satisfied, since $1 + 4 = 2 + 3$.
Next, assume that for some natural number $s$, there exists such a partition of the set $\{1, 2... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 879 |
VII OM - I - Task 3
In a square $ABCD$ with area $S$, vertex $A$ is connected to the midpoint of side $BG$, vertex $B$ is connected to the midpoint of side $CD$, vertex $C$ is connected to the midpoint of side $DA$, and vertex $D$ is connected to the midpoint of side $AB$. Calculate the area of the part of the square ... | The solution to the problem is illustrated in Fig. 1, where $ M $, $ N $, $ P $, $ Q $ denote the midpoints of the sides, and $ O $ the center of the square; through the vertices $ A $, $ B $, $ C $, $ D $ lines parallel to the lines $ DQ $, $ AM $, $ BN $, $ CP $ have been drawn. If the figure is rotated by $ 90^\circ... | \frac | Geometry | math-word-problem | Yes | Yes | olympiads | false | 880 |
VI OM - I - Problem 10
Prove that the number $ 53^{53} - 33^{33} $ is divisible by $ 10 $. | It is enough to prove that the last digits of the numbers $53^{53}$ and $33^{33}$ are the same. The last digit of the product $(10k + a)(10m + b)(10n + c) \ldots$, where $k, m, n, \ldots a, b, c, \ldots$ are natural numbers, is equal to the last digit of the product $abc\ldots$. The number $53^{53}$ therefore has the s... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 881 |
XIX OM - I - Problem 6
Prove that if the polynomial $ F(x) $ with integer coefficients has a root $ \frac{p}{q} $, where $ p $ and $ q $ are relatively prime integers, then $ p - q $ is a divisor of the number $ F(1) $, and $ p + q $ is a divisor of the number $ F(-1) $. | Let us substitute $ x $ with the variable $ y = qx $. Equation (1) then takes the form
Consider the polynomial
whose coefficients are integers. It has an integer root $ y=p $, hence
where $ Q(y) $ is a polynomial with integer coefficients*).
Substituting $ y = q $ into this equality, we get
The numbers $ q^n $ and... | proof | Algebra | proof | Yes | Yes | olympiads | false | 882 |
XLVII OM - I - Problem 5
In the plane, a triangle $ ABC $ is given, where $ |\measuredangle CAB| = \alpha > 90^{\circ} $, and a segment $ PQ $, whose midpoint is point $ A $. Prove that | Let $ M $ be the midpoint of side $ BC $ of triangle $ ABC $, and $ O $ - the center of the circle $ \omega $ circumscribed around this triangle. Angle $ CAB $ is obtuse by assumption. It follows that points $ A $ and $ O $ lie on opposite sides of line $ BC $; in particular, point $ O $ does not coincide with $ M $.
L... | proof | Geometry | proof | Yes | Yes | olympiads | false | 883 |
XXIII OM - III - Problem 4
On a straight line without common points with the sphere $K$, points $A$ and $B$ are given. The orthogonal projection $P$ of the center of the sphere $K$ onto the line $AB$ lies between points $A$ and $B$, and $AP$ and $BP$ are greater than the radius of the sphere. We consider the set $Z$ o... | We will first prove the
Lemma. If $ S $ is a vertex, $ O $ is the center of the base of a right circular cone, and point $ T $ lies in the plane of the base, then the line $ ST $ forms the largest angle with such a generatrix $ SR $ of the cone that $ O \in \overline{TR} $ (Fig. 16).
Proof. For any point $ R $ on the c... | proof | Geometry | proof | Yes | Yes | olympiads | false | 885 |
X OM - III - Task 5
In the plane of triangle $ ABC $, a line moves, intersecting sides $ AC $ and $ BC $ at points $ D $ and $ E $ such that $ AD = BE $. Find the geometric locus of the midpoint $ M $ of segment $ DE $. | When $ AC = BC $, the sought geometric locus is the angle bisector of $ C $. It remains to consider only the case where $ AC \ne BC $. We will further assume that $ AC < BC $.
First, note that when $ AC < BC $, the line $ DE $ intersects the line $ AB $ at a point lying on the extension of $ AB $ beyond point $ B $, si... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 886 |
LIV OM - II - Task 4
Prove that for every prime number $ p > 3 $ there exist integers $ x $, $ y $, $ k $ satisfying the conditions: $ 0 < 2k < p $ and | Let $ A $ be the set of numbers of the form $ x^2 $, where $ 0 \leq x < p/2 $, and $ B $ the set of numbers of the form $ 3 - y^2 $, where $ 0 \leq y < p/2 $; the numbers $ x $, $ y $ are integers. We will show that the numbers in the set $ A $ give different remainders when divided by $ p $.
Suppose that the numbers $... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 889 |
XLVIII OM - III - Problem 2
Find all triples of real numbers $ x $, $ y $, $ z $ satisfying the system of equations | Suppose that the numbers $ x $, $ y $, $ z $ satisfy the given system; from the second equation it follows that
Further, we have
which, by the first equation of the system, gives:
There is also the inequality
that is,
By property (1), we can multiply inequalities (3) and (4) side by side:
... | (x,y,z)=(\1/3,\1/3,\1/3)(x,y,z)=(\1/\sqrt{3},0,0),(0,\1/\sqrt{3},0),(0,0,\1/\sqrt{3}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 890 |
L OM - I - Task 3
In an isosceles triangle $ ABC $, angle $ BAC $ is a right angle. Point $ D $ lies on side $ BC $, such that $ BD = 2 \cdot CD $. Point $ E $ is the orthogonal projection of point $ B $ onto line $ AD $. Determine the measure of angle $ CED $. | Let's complete the triangle $ABC$ to a square $ABFC$. Assume that line $AD$ intersects side $CF$ at point $P$, and line $BE$ intersects side $AC$ at point $Q$. Since
$ CP= \frac{1}{2} CF $. Using the perpendicularity of lines $AP$ and $BQ$ and the above equality, we get $ CQ= \frac{1}{2} AC $, and consequently $... | 45 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 891 |
XXI OM - III - Task 4
In the plane, there are $ n $ rectangles with sides respectively parallel to two given perpendicular lines. Prove that if any two of these rectangles have at least one point in common, then there exists a point belonging to all the rectangles. | Let us choose a simple set of axes for the coordinate system (Fig. 14). Then the vertices of the $i$-th rectangle $P_i$ will have coordinates $(x_i, y_i)$, $(x_i, y)$, $(x, y_i)$, $(x, y)$, where $x_i < x$, $y_i < y$. The point $(x, y)$ will therefore belong to the rectangle $P_i$ if and only if $x_i \leq x \leq x$ and... | proof | Geometry | proof | Yes | Yes | olympiads | false | 895 |
XXXVIII OM - I - Problem 4
For a given natural number $ n $, we consider the family $ W $ of all sets $ A $ of pairs of natural numbers not greater than $ n $, such that if $ (a, b) \in A $, then $ (x, y) \in A $ for $ 1 \leq x \leq a $, $ 1 \leq y \leq b $.
Prove that the family $ W $ consists of $ \binom{2n}{n} $ di... | For any pair of natural numbers $ (x,y) $, we denote by $ Q_{xy} $ the unit square with vertices $ (x-1,y-1) $, $ (x,y-1) $, $ (x,y) $, $ (x-1,y) $, and by $ R_{xy} $ - the rectangle with vertices $ (0,0) $, $ (x,0) $, $ (x,y) $, $ (0,y) $. If $ A $ is any set of pairs of natural numbers, then the symbol $ Z(A) $ will ... | \binom{2n}{n} | Combinatorics | proof | Yes | Yes | olympiads | false | 897 |
XXIV OM - I - Problem 11
Prove that if the center of the sphere circumscribed around a tetrahedron coincides with the center of the sphere inscribed in this tetrahedron, then the faces of this tetrahedron are congruent. | Let point $O$ be the center of the inscribed and circumscribed spheres of the tetrahedron $ABCD$. Denote by $r$ and $R$ the radii of these spheres, respectively. If $O$ is the point of tangency of the inscribed sphere with one of the faces, and $P$ is one of the vertices of this face, then applying the Pythagorean theo... | proof | Geometry | proof | Yes | Yes | olympiads | false | 900 |
VIII OM - I - Problem 7
Construct triangle $ ABC $ given: side $ AB = c $, circumradius $ R $, and angle $ \delta $ between the angle bisector of $ C $ and the altitude from vertex $ C $. | Let $D$ (Fig. 4) denote the foot of the altitude of triangle $ABC$ drawn from vertex $C$, $E$ - the point of intersection of the angle bisector of angle $C$ with side $AB$, $M$ - the point of intersection of line $CE$ with the circumcircle of triangle $ABC$, and $F$ - the foot of the perpendicular dropped from $M$ to l... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 901 |
II OM - II - Task 6
Given points $ A $ and $ B $ and a circle $ k $. Construct a circle passing through points $ A $ and $ B $ and intersecting circle $ k $ in a common chord of given length $ d $. | Analyze. The task boils down to determining the center $X$ of the common chord of the given circle and the circle to be found; for, having the point $X$, we can draw in the given circle $k$ a chord $CD$ with the center $X$ and draw a circle through the points $A$, $B$, $C$, $D$.
The geometric locus of the centers $X$ o... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 902 |
XXX OM - I - Task 8
Among the cones inscribed in the sphere $ B $, the cone $ S_1 $ was chosen such that the sphere $ K_1 $ inscribed in the cone $ S_1 $ has the maximum volume. Then, a cone $ S_2 $ of maximum volume was inscribed in the sphere $ B $, and a sphere $ K_2 $ was inscribed in the cone $ S_2 $. Determine w... | Let $ S $ be any cone inscribed in a given sphere $ B $, and $ K $ - the sphere inscribed in the cone $ S $. The axial section of the cone $ S $ is an isosceles triangle $ ACD $, where $ A $ is the vertex of the cone (Fig. 7).
om30_1r_img_7.jpg
Let $ O $ be the center of the sphere $ B $, and $ Q $ - the center of the ... | V(\frac{\sqrt{3}}{3})>V(\frac{1}{2}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 904 |
XXXI - I - Task 1
Determine for which values of the parameter $ a $ a rhombus with side length $ a $ is a cross-section of a cube with edge length 2 by a plane passing through the center of the cube. | Suppose the rhombus $KLMN$ is a cross-section of the cube $ABCDA_1B_1C_1D_1$ by a plane passing through the center of the cube $O$. Since $O$ is also the center of symmetry of the cross-section, the opposite vertices of the rhombus lie on opposite edges of the cube. Let, for example, as in the figure, $K \in \overline{... | [2,\sqrt{5}] | Geometry | math-word-problem | Yes | Yes | olympiads | false | 905 |
XIV OM - II - Task 6
From point $ S $ in space, $ 3 $ rays emerge: $ SA $, $ SB $, and $ SC $, none of which is perpendicular to the other two. Through each of these rays, a plane is drawn perpendicular to the plane containing the other two rays. Prove that the planes drawn intersect along a single line $ d $. | In the case where the half-lines lie in the same plane, the theorem is obvious; the line $d$ is then perpendicular to the given plane. Let us assume, therefore, that the half-lines $SA$, $SB$, $SC$ form a trihedral angle. According to the assumption, at most one of the dihedral angles of this trihedron is a right angle... | proof | Geometry | proof | Yes | Yes | olympiads | false | 906 |
VIII OM - I - Zadanie 5
Jaki warunek powinna spełniać liczbą $ q $, aby istniał trójkąt, którego boki tworzą postęp geometryczny o ilorazie $ q $?
|
Weźmy pod uwagę trzy odcinki, których długości tworzą postęp geometryczny o ilorazie $ q $. Ponieważ stosunek dwóch odcinków jest liczbą niezależną od jednostki, jaką te odcinki mierzymy, więc jeśli za jednostkę obierzemy pierwszy z rozważanych trzech odcinków, to długości tych odcinków wyrażą się liczbami $ 1 $, $ q ... | \frac{-1+\sqrt{5}}{2}<\frac{1+\sqrt{5}}{2} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 912 |
XL OM - III - Task 5
On a sphere of radius $ r $, there are three circles of radius $ a $, each tangent to the other two and contained within the same hemisphere. Find the radius of the circle lying on the same sphere and tangent to each of these three circles.
Note: Similar to on a plane, we say that two circles lyin... | Let's denote three given circles (of radius $a$) by $k_1$, $k_2$, $k_3$. Suppose $k_0$ is a circle of radius $x$, located on the same sphere and tangent to each of the circles $k_1$, $k_2$, $k_3$. None of the circles $k_1$, $k_2$, $k_3$ are great circles of the given sphere (two different great circles always have two ... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 915 | |
XXII OM - II - Problem 4
In the plane, a finite set of points $ Z $ is given with the property that no two distances between points in $ Z $ are equal. Points $ A, B $ belonging to $ Z $ are connected if and only if $ A $ is the nearest point to $ B $ or $ B $ is the nearest point to $ A $. Prove that no point in the ... | Suppose that point $ A \in Z $ is connected to some points $ B $ and $ C $ of set $ Z $ and let, for example, $ AB < AC $. Then point $ C $ is not the closest to $ A $, and from the conditions of the problem it follows that point $ A $ is the closest to $ C $. Therefore, $ AC < BC $. Hence, in triangle $ ABC $, side $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 917 |
LII OM - III - Problem 6
Given positive integers $ n_1 < n_2 < \ldots < n_{2000} < 10^{100} $. Prove that from the set $ \{n_1, n_2, \ldots, n_{2000}\} $, one can select non-empty, disjoint subsets $ A $ and $ B $ having the same number of elements, the same sum of elements, and the same sum of squares of elements. | For a set $ X \subseteq \{n_1,n_2,\ldots,n_{2000}\} $, let $ s_0(X) $, $ s_1(X) $, and $ s_2(X) $ denote the number of elements, the sum of elements, and the sum of squares of elements of the set $ X $, respectively.
It suffices to prove that there exist two different subsets $ C $ and $ D $ of the set $ \{n_1,n_2,\ldo... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 920 |
XLVIII OM - II - Problem 6
In a cube with edge length $1$, there are eight points. Prove that some two of them are the endpoints of a segment of length not greater than $1$. | Let the vertices of a given cube $\mathcal{C}$ be denoted by $A_1, \ldots, A_8$. Let $\mathcal{C}_i$ be the cube with edge length $\frac{1}{2}$, having one vertex at point $A_i$ and three faces contained in the faces of cube $\mathcal{C}$. Let $P_1, \ldots, P_8$ be eight given points.
Each cube $\mathcal{C}_i$ has a di... | proof | Geometry | proof | Yes | Yes | olympiads | false | 921 |
L OM - I - Problem 1
Prove that among numbers of the form $ 50^n + (50n + 1)^{50} $, where $ n $ is a natural number, there are infinitely many composite numbers. | I way:
If $ n $ is an odd number, then the number $ 50^n $ when divided by $ 3 $ gives a remainder of $ 2 $. If, moreover, $ n $ is divisible by $ 3 $, then $ (50n+1)^{50} $ when divided by $ 3 $ gives a remainder of $ 1 $. Hence, the number $ 50^n + (50n+1)^{50} $ is divisible by $ 3 $ for numbers $ n $ of the form $... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 922 |
XXXIII OM - II - Problem 2
The plane is covered with circles in such a way that the center of each of these circles does not belong to any other circle. Prove that each point of the plane belongs to at most five circles. | Suppose that point $ A $ belongs to six circles from the considered family and that $ O_1 $, $ O_2 $, $ O_3 $, $ O_4 $, $ O_5 $, $ O_6 $ are the centers of these circles. It follows that one of the angles $ O_iAO_j $ $ (i,j = 1, 2, \ldots, 6) $ has a measure not greater than $ 60^\circ $ (six of the angles $ O_iAO_j $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 923 |
LVII OM - II - Problem 5
Point $ C $ is the midpoint of segment $ AB $. Circle $ o_1 $ passing through points $ A $ and $ C $ intersects circle $ o_2 $ passing
through points $ B $ and $ C $ at different points $ C $ and $ D $. Point $ P $ is the midpoint of the arc $ AD $ of circle $ o_1 $, which does not
contain poi... | If $ AC = CD $, then also $ BC = CD $. Then segments $ PC $ and $ QC $ are diameters of circles $ o_1 $ and $ o_2 $, respectively. Therefore, $ \measuredangle CDP = \measuredangle CDQ = 90^{\circ} $, which implies that point $ D $ lies on segment $ PQ $, and lines $ PQ $ and $ CD $ are perpendicular.
Let us assume in t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 924 |
XVIII OM - II - Task 2
In a room, there are 100 people, each of whom knows at least 66 of the remaining 99 people. Prove that it is possible that in every quartet of these people, there are two who do not know each other. We assume that if person $ A $ knows person $ B $, then person $ B $ also knows person $ A $. | Let's denote the people in the room by the letters $A_1, A_2, \ldots, A_{100}$. Let $M$ be the set of people $\{A_1, A_2, \ldots, A_{33}\}$, $N$ be the set of people $\{A_{34}, A_{35}, \ldots, A_{66}\}$, and $P$ be the set of people $\{A_{67}, A_{68}, \ldots, A_{100}\}$. The case described in the problem occurs, for ex... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 925 |
LV OM - I - Task 7
Find all solutions to the equation $ a^2+b^2=c^2 $ in positive integers such that the numbers $ a $ and $ c $ are prime, and the number $ b $ is the product of at most four prime numbers. | There are three solutions $(a,b,c): (3,4,5), (5,12,13), (11,60,61)$. Let $a$, $b$, $c$ be numbers satisfying the conditions of the problem. Then
from the assumption that the number $a$ is prime, we obtain $c = b + 1$. Therefore, $a^2 = 2b + 1$, which implies that the number $a$ is odd.
Let $a = 2n + 1$. Then
if the n... | (3,4,5),(5,12,13),(11,60,61) | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 926 |
XLV OM - III - Task 2
In the plane, there are two parallel lines $ k $ and $ l $ and a circle disjoint from line $ k $. From a point $ A $ lying on line $ k $, we draw two tangents to this circle intersecting line $ l $ at points $ B $, $ C $. Let $ m $ be the line passing through point $ A $ and the midpoint of segme... | Note 1. The conditions of the problem do not change if the given line $ l $ is replaced by another line $ l' $ parallel to $ k $ and different from $ k $. Indeed: if two tangents to a given circle emanating from point $ A $ intersect the line $ l $ at points $ B $ and $ C $, then the midpoints of segments $ BC $ and $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 927 |
XIV OM - II - Task 4
In triangle $ABC$, the angle bisectors of the internal and external angles at vertices $A$ and $B$ have been drawn. Prove that the perpendicular projections of point $C$ onto these bisectors lie on the same line. | Let $M$ and $N$ denote the projections of point $C$ onto the angle bisector $AM$ of angle $A$ of triangle $ABC$ and onto the bisector of the adjacent angle (Fig. 22).
Since the bisectors $AM$ and $AN$ are perpendicular, the quadrilateral $AMCN$ is a rectangle. The point $S$, where the diagonals $MN$ and $AC$ intersect,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 928 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.